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Li, Chao; Zhu, Yihang
W. Zhang’s arithmetic fundamental lemma (AFL) is a conjectural identity between the derivative of an orbital integral on a symmetric space with an arithmetic intersection number on a unitary Rapoport–Zink space. In the minuscule case, Rapoport, Terstiege and Zhang have verified the AFL conjecture via explicit evaluation of both sides of the identity. We present a simpler way for evaluating the arithmetic intersection number, thereby providing a new proof of the AFL conjecture in the minuscule case.

Patrikis, Stefan
In previous work, we described conditions under which a single geometric representation [math] of the Galois group of a number field [math] lifts through a central torus quotient [math] to a geometric representation. In this paper, we prove a much sharper result for systems of [math] -adic representations, such as the [math] -adic realizations of a motive over [math] , having common “good reduction” properties. Namely, such systems admit geometric lifts with good reduction outside a common finite set of primes. The method yields new proofs of theorems of Tate (the original result on lifting projective representations over number fields)...

Pizzato, Marco; Sano, Taro; Tasin, Luca
We show that the Ambro–Kawamata nonvanishing conjecture holds true for a quasismooth WCI [math] which is Fano or Calabi–Yau, i.e., we prove that, if [math] is an ample Cartier divisor on [math] , then [math] is not empty. If [math] is smooth, we further show that the general element of [math] is smooth. We then verify the Ambro–Kawamata conjecture for any quasismooth weighted hypersurface. We also verify Fujita’s freeness conjecture for a Gorenstein quasismooth weighted hypersurface. ¶ For the proofs, we introduce the arithmetic notion of regular pairs and highlight some interesting connections with the Frobenius coin problem.

Castella, Francesc; Kim, Chan-Ho; Longo, Matteo
Building on the construction of big Heegner points in the quaternionic setting by Longo and Vigni, and their relation to special values of Rankin–Selberg [math] -functions established by Castella and Longo, we obtain anticyclotomic analogues of the results of Emerton, Pollack and Weston on the variation of Iwasawa invariants in Hida families. In particular, combined with the known cases of the anticyclotomic Iwasawa main conjecture in weight [math] , our results yield a proof of the main conjecture for [math] -ordinary newforms of higher weights and trivial nebentypus.

Ru, Min; Wang, Julie Tzu-Yueh
We establish a height inequality, in terms of an (ample) line bundle, for a sum of subschemes located in [math] -subgeneral position in an algebraic variety, which extends a result of McKinnon and Roth (2015). The inequality obtained in this paper connects the result of McKinnon and Roth (the case when the subschemes are points) and the results of Corvaja and Zannier (2004), Evertse and Ferretti (2008), Ru (2017), and Ru and Vojta (2016) (the case when the subschemes are divisors). Furthermore, our approach gives an alternative short and simpler proof of McKinnon and Roth’s result.

Blasius, Don; Guerberoff, Lucio
In this paper we study the action of complex conjugation on Shimura varieties and the problem of descending Shimura varieties to the maximal totally real field of the reflex field. We prove the existence of such a descent for many Shimura varieties whose associated adjoint group has certain factors of type [math] or [math] . This includes a large family of Shimura varieties of abelian type. Our considerations and constructions are carried out purely at the level of Shimura data and group theory.

Helm, David; Tian, Yichao; Xiao, Liang
Let [math] be a real quadratic field in which a fixed prime [math] is inert, and [math] be an imaginary quadratic field in which [math] splits; put [math] . Let [math] be the fiber over [math] of the Shimura variety for [math] with hyperspecial level structure at [math] for some integer [math] . We show that under some genericity conditions the middle-dimensional Tate classes of [math] are generated by the irreducible components of its supersingular locus. We also discuss a general conjecture regarding special cycles on the special fibers of unitary Shimura varieties, and on their relation to Newton stratification.

Borger, James
We give a concrete description of the category of étale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical [math] -typical and big Witt vector functors but also for certain analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for classical Witt vectors. Our larger purpose is to provide the affine foundations for the algebraic geometry...

Borger, James
We give a concrete description of the category of étale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical [math] -typical and big Witt vector functors but also for certain analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for classical Witt vectors. Our larger purpose is to provide the affine foundations for the algebraic geometry...

Stange, Katherine
An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose [math] is an elliptic curve over a field [math] , and [math] are points on [math] defined over [math] . To this information we associate an [math] -dimensional array of values in [math] satisfying a nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. We demonstrate an explicit bijection between the set of elliptic nets and the set of elliptic...

Stange, Katherine
An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose [math] is an elliptic curve over a field [math] , and [math] are points on [math] defined over [math] . To this information we associate an [math] -dimensional array of values in [math] satisfying a nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. We demonstrate an explicit bijection between the set of elliptic nets and the set of elliptic...

Lazarsfeld, Robert; Pareschi, Giuseppe; Popa, Mihnea
We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang and To on projective normality.

Lazarsfeld, Robert; Pareschi, Giuseppe; Popa, Mihnea
We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang and To on projective normality.

Saïdi, Mohamed; Tamagawa, Akio
We prove that a certain class of open homomorphisms between Galois groups of function fields of curves over finite fields arises from embeddings between the function fields.

Saïdi, Mohamed; Tamagawa, Akio
We prove that a certain class of open homomorphisms between Galois groups of function fields of curves over finite fields arises from embeddings between the function fields.

Brüstle, Thomas; Zhang, Jie
We study the cluster category [math] of a marked surface [math] without punctures. We explicitly describe the objects in [math] as direct sums of homotopy classes of curves in [math] and one-parameter families related to noncontractible closed curves in [math] . Moreover, we describe the Auslander–Reiten structure of the category [math] in geometric terms and show that the objects without self-extensions in [math] correspond to curves in [math] without self-intersections. As a consequence, we establish that every rigid indecomposable object is reachable from an initial triangulation.

Bröker, Reinier; Gruenewald, David; Lauter, Kristin
For a complex abelian surface [math] with endomorphism ring isomorphic to the maximal order in a quartic CM field [math] , the Igusa invariants [math] generate an unramified abelian extension of the reflex field of [math] . In this paper we give an explicit geometric description of the Galois action of the class group of this reflex field on [math] . Our description can be expressed by maps between various Siegel modular varieties, and we can explicitly compute the action for ideals of small norm. We use the Galois action to modify the CRT method for computing Igusa class polynomials,...

Ingram, Patrick
Let [math] be an elliptic surface defined over a number field [math] , let [math] be a section, and let [math] be a rational prime. We bound the number of points of low algebraic degree in the [math] -division hull of [math] at the fibre [math] . Specifically, for [math] with [math] such that [math] is nonsingular, we obtain a bound on the number of [math] such that [math] , and such that [math] for some [math] . This bound depends on [math] , [math] , [math] , [math] , and [math] , but is independent of [math] .